# American Institute of Mathematical Sciences

September 2018, 17(5): 1723-1747. doi: 10.3934/cpaa.2018082

## Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities

 School of Mathematics and Information Science, Guangzhou University, Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

* Corresponding author

Received  March 2017 Revised  November 2017 Published  April 2018

By using critical point theory, we obtain some new sufficient conditions on the existence of homoclinic solutions of a class of nonlinear discrete $\phi$-Laplacian equations with mixed nonlinearities for the potentials being periodic or being unbounded, respectively. And we prove it is also necessary in some special cases. In addition, multiplicity results of homoclinic solutions for nonlinear discrete $\phi$-Laplacian equations with unbounded potentials have also been considered. In our paper, the nonlinearities can be mixed super $p$-linear with asymptotically $p$-linear at $∞$ for $p≥ 1$. To the best of our knowledge, there is no such result for the existence of homoclinic solutions with discrete $\phi$-Laplacian before. Finally, an extension has also been considered.

Citation: Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082
##### References:
 [1] G. Arioli and F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26 (1996), 1103-1114. [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250. [3] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Physica D, 216 (2006), 1-30. [4] G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496-5507. [5] G. Chen and S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232 (2014), 787-798. [6] W. Chen and M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 351-360. [7] J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76. [8] S. Flach and A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 467 (2008), 1-116. [9] J. W. Fleischer, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), 147-150. [10] A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29 (2004), 77-93. [11] G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63. [12] A. Khare, K. Rasmussen, M. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814. [13] G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), Art. ID 165501. [14] W. Krolikowski, B. L. Davies and C. Denz, Photorefractive solitons, IEEE J. Quant. Electron., 39 (2003), 3-12. [15] J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), 208-218. [16] G. Lin and Z. Zhou, Periodic and subharmonic solutions for a $2n$th-order difference equation containing both advance and retardation with $\phi$-Laplacian, Adv. Difference Equ., 2014(2014), Art. ID 74. [17] G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Method Appl. Sci., 39 (2016), 245-260. [18] G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. lett., 64 (2017), 15-20. [19] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica Chin. Ser., 46 (2003), 625-630. [20] R. Livi, R. Franzosi and G. L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), Art. ID 060401. [21] S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64 (2013), 1413-1442. [22] A. Mai and Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), Art. ID 317139. [23] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672-4687. [24] J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065-1076. [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265. [26] A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 464 (2008), 3219-3236. [27] H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065-1075. [28] H. Shi and H. Zhang, Existence of gap solitons in a periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419. [29] C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 2-4 (2011), 569-588. [30] A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28 (2003), 2345-2347. [31] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32 (2016), 463-473. [32] X. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Adv. Difference Equ., 2013 (2013), Art. ID 242. [33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [34] M. Yang, W. Chen and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum $0$, Acta. Appl. Math., 110 (2010), 1475-1488. [35] G. Zhang and A. Pankov, Standing waves of the discrete nonlinear Schrödinger equations with growing potentials, Commun. Math. Anal., 5 (2008), 38-49. [36] Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58 (2015), 781-790. [37] Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212. [38] Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 1809-1822. [39] Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic diffrence equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93. [40] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.

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##### References:
 [1] G. Arioli and F. Gazzola, Periodic motions of an infinite lattice of particles with nearest neighbor interaction, Nonlinear Anal., 26 (1996), 1103-1114. [2] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201-250. [3] S. Aubry, Discrete breathers: localization and transfer of energy in discrete Hamiltonian nonlinear systems, Physica D, 216 (2006), 1-30. [4] G. Chen and S. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496-5507. [5] G. Chen and S. Ma, Homoclinic solutions of discrete nonlinear Schrödinger equations with asymptotically or super linear terms, Appl. Math. Comput., 232 (2014), 787-798. [6] W. Chen and M. Yang, Standing waves for periodic discrete nonlinear Schrödinger equations with asymptotically linear terms, Acta Math. Appl. Sin. Engl. Ser., 28 (2012), 351-360. [7] J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity, Physica D, 238 (2009), 67-76. [8] S. Flach and A. V. Gorbach, Discrete breathers-advance in theory and applications, Phys. Rep., 467 (2008), 1-116. [9] J. W. Fleischer, M. Segev, N. K. Efremidis and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), 147-150. [10] A. V. Gorbach and M. Johansson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model, Eur. Phys. J. D, 29 (2004), 77-93. [11] G. James, Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13 (2003), 27-63. [12] A. Khare, K. Rasmussen, M. Samuelsen and A. Saxena, Exact solutions of the saturable discrete nonlinear Schrödinger equation, J. Phys. A, 38 (2005), 807-814. [13] G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), Art. ID 165501. [14] W. Krolikowski, B. L. Davies and C. Denz, Photorefractive solitons, IEEE J. Quant. Electron., 39 (2003), 3-12. [15] J. Kuang and Z. Guo, Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities, Nonlinear Anal., 89 (2013), 208-218. [16] G. Lin and Z. Zhou, Periodic and subharmonic solutions for a $2n$th-order difference equation containing both advance and retardation with $\phi$-Laplacian, Adv. Difference Equ., 2014(2014), Art. ID 74. [17] G. Lin and Z. Zhou, Homoclinic solutions in periodic difference equations with mixed nonlinearities, Math. Method Appl. Sci., 39 (2016), 245-260. [18] G. Lin and Z. Zhou, Homoclinic solutions in non-periodic discrete $\phi$-Laplacian equations with mixed nonlinearities, Appl. Math. lett., 64 (2017), 15-20. [19] S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica Chin. Ser., 46 (2003), 625-630. [20] R. Livi, R. Franzosi and G. L. Oppo, Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), Art. ID 060401. [21] S. Ma and Z. Wang, Multibump solutions for discrete periodic nonlinear Schrödinger equations, Z. Angew. Math. Phys., 64 (2013), 1413-1442. [22] A. Mai and Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), Art. ID 317139. [23] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Anal., 75 (2012), 4672-4687. [24] J. Mawhin, Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1065-1076. [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, J. Math. Anal. Appl., 371 (2010), 254-265. [26] A. Pankov and V. Rothos, Periodic and decaying solutions in discrete nonlinear Schrödinger with saturable nonlinearity, Proc. R. Soc. A, 464 (2008), 3219-3236. [27] H. Shi, Gap solitons in periodic discrete Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065-1075. [28] H. Shi and H. Zhang, Existence of gap solitons in a periodic discrete nonlinear Schrödinger equations, J. Math. Anal. Appl., 361 (2010), 411-419. [29] C. A. Stuart, Locating Cerami sequences in a mountain pass geometry, Commun. Appl. Anal., 2-4 (2011), 569-588. [30] A. A. Sukhorukov and Y. S. Kivshar, Generation and stability of discrete gap solitons, Opt. Lett., 28 (2003), 2345-2347. [31] X. Tang, Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation, Acta Math. Sin. Engl. Ser., 32 (2016), 463-473. [32] X. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Adv. Difference Equ., 2013 (2013), Art. ID 242. [33] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [34] M. Yang, W. Chen and Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum $0$, Acta. Appl. Math., 110 (2010), 1475-1488. [35] G. Zhang and A. Pankov, Standing waves of the discrete nonlinear Schrödinger equations with growing potentials, Commun. Math. Anal., 5 (2008), 38-49. [36] Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math., 58 (2015), 781-790. [37] Z. Zhou and J. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212. [38] Z. Zhou and J. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 1809-1822. [39] Z. Zhou, J. Yu and Y. Chen, Homoclinic solutions in periodic diffrence equations with saturable nonlinearity, Sci. China Math., 54 (2011), 83-93. [40] W. Zou, Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358.
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